A361044 - OEIS

%I #11 Mar 18 2023 08:49:14

%S 1,3,8,12,8,3,1,1,5,24,60,109,161,161,109,60,24,5,1,1,7,48,168,483,

%T 1074,1805,2531,2886,2531,1805,1074,483,168,48,7,1

%N Triangle read by rows. T(n, k) is the k-th Lie-Betti number of the friendship (or windmill) graph, for n >= 1.

%C The triangle is inspired by _Samuel J. Bevins_'s A360571.

%C The friendship graph is constructed by joining n copies of the cycle graph C_3 at a common vertex. F_1 is isomorphic to C_3 (the triangle graph) and has 3 vertices, F_2 is the butterfly graph and has 5 vertices and if n > 2 then F_n has 2*n + 1 vertices.

%H Marco Aldi and Samuel Bevins, <a href="https://arxiv.org/abs/2212.13608">L_oo-algebras and hypergraphs</a>, arXiv:2212.13608 [math.CO], 2022. See page 9.

%H Meera G. Mainkar, <a href="https://arxiv.org/abs/1310.3414">Graphs and two step nilpotent Lie algebras</a>, arXiv:1310.3414 [math.DG], 2013. See page 1.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DutchWindmillGraph.html">Dutch Windmill Graph</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Friendship_graph">Friendship Graph</a>.

%e The triangle T(n, k) starts:

%e [1] 1, 3, 8, 12, 8, 3, 1;

%e [2] 1, 5, 24, 60, 109, 161, 161, 109, 60, 24, 5, 1;

%e [3] 1, 7, 48, 168, 483, 1074, 1805, 2531, 2886, 2531, 1805, 1074, 483, 168, 48, 7, 1;

%o (SageMath)

%o from sage.algebras.lie_algebras.lie_algebra import LieAlgebra, LieAlgebras

%o def BettiNumbers(graph):

%o     D = {}

%o     for edge in graph.edges():

%o         e = "x" + str(edge[0])

%o         f = "x" + str(edge[1])

%o         D[(e, f)] = {e + f : 1}

%o     C = (LieAlgebras(QQ).WithBasis().Graded().FiniteDimensional().

%o          Stratified().Nilpotent())

%o     L = LieAlgebra(QQ, D, nilpotent=True, category=C)

%o     H = L.cohomology()

%o     d = L.dimension() + 1

%o     return [H[n].dimension() for n in range(d)]

%o def A361044_row(n):

%o     return BettiNumbers(graphs.FriendshipGraph(n))

%o for n in range(1, 4): print(A361044_row(n))

%Y Cf. A360571 (path graph), A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A360937 (wheel graph).

%K nonn,tabf,more

%O 1,2

%A _Peter Luschny_, Mar 01 2023